I've been told to solder the bottom, then use a knife to move up the plastic part, then solder the top, then move the plastic back down. The problem is that the small abount of solder on the pins makes it nearly impossible to move the plastic back down without damaging anything.

Hmm... Well, always try not to apply any forces to the soldered joint. So, after you soldered the bottom, turn the board upside down, close to the edge of the table, so it stays on the pins you just soldered. Then carefully insert the blade at the base of the pins and push down towards the table so you slide that plastic down the pins. After you solder the top, press on the plastic while the pins rest on the table, so you don't force the soldered joints.

I would try to slide the plastic on the pins before you plug them into the board, to loosen up the plastic and ease the process. Also you can try to make a jig from static foam (the thing the ICs are plugged when you buy them) and have the plastic already moved in the middle of the pins before you solder the bottom.

Er... any reason why you should do this in the first place? Maybe for homemade double-sided PCBs that don't plate through, but for production PCBs you won't need to do that. A proper header pin solder joint on the bottom will actually flow through the hole and a little will get towards the top, which should be plenty